Determine the values of a to e in order giving, reasons for all your statements.

The angle $45^\circ$ and the angle $y+20^\circ$ are alternate interior angles.

The angle $x$ and the $45^\circ$ angle are consecutive interior angles (also known as co-interior angles).

The values are: $\boxed{x = 135^\circ, y = 25^\circ}$

The two sides marked with dashes are equal, so the triangle is isosceles. The angles opposite these sides are equal. Let the angle at the bottom right vertex inside the triangle be $A$.

The sum of angles in a triangle is $180^\circ$.

Substitute $A=x$:

So, $A = 78^\circ$.

The exterior angle $2y$ is equal to the sum of the two opposite interior angles ($24^\circ$ and $x$).

Substitute $x = 78^\circ$:

The values are: $\boxed{x = 78^\circ, y = 51^\circ}$

3. Determine the value of $x, y$ and $z$ in the following diagram, showing all working and giving reasons for each statement that you make.

The line segment AD is parallel to the line segment BE (indicated by arrows). The transversal is AB.
The angle $3x-10^\circ$ and the angle $58^\circ$ are alternate interior angles.

The line segment AD is parallel to the line segment BE. The transversal is DE.
The angle $58^\circ$ and the angle $y$ are alternate interior angles.

The angles in triangle BCE are $z$, $42^\circ$, and the angle at C.
The angle at C (angle BCE) and angle $y$ are vertically opposite angles.
So, angle BCE = $y = 58^\circ$.
Now, sum of angles in triangle BCE:

The values are: $\boxed{x = 22.67^\circ, y = 58^\circ, z = 80^\circ}$